Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics

We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, this small window converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice QCD applications, where hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides. Deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.Comment: 22 pages, 26 eps figure

Vibration stimuli and the differentiation of musculoskeletal progenitor cells: Review of results in vitro and in vivo

Due to the increasing burden on healthcare budgets of musculoskeletal system disease and injury, there is a growing need for safe, effective and simple therapies. Conditions such as osteoporosis severely impact on quality of life and result in hundreds of hours of hospital time and resources. There is growing interest in the use of low magnitude, high frequency vibration (LMHFV) to improve bone structure and muscle performance in a variety of different patient groups. The technique has shown promise in a number of different diseases, but is poorly understood in terms of the mechanism of action. Scientific papers concerning both the in vivo and in vitro use of LMHFV are growing fast, but they cover a wide range of study types, outcomes measured and regimens tested. This paper aims to provide an overview of some effects of LMHFV found during in vivo studies. Furthermore we will review research concerning the effects of vibration on the cellular responses, in particular for cells within the musculoskeletal system. This includes both osteogenesis and adipogenesis, as well as the interaction between MSCs and other cell types within bone tissue

Towards Real-Time Detection and Tracking of Spatio-Temporal Features: Blob-Filaments in Fusion Plasma

A novel algorithm and implementation of real-time identification and tracking of blob-filaments in fusion reactor data is presented. Similar spatio-temporal features are important in many other applications, for example, ignition kernels in combustion and tumor cells in a medical image. This work presents an approach for extracting these features by dividing the overall task into three steps: local identification of feature cells, grouping feature cells into extended feature, and tracking movement of feature through overlapping in space. Through our extensive work in parallelization, we demonstrate that this approach can effectively make use of a large number of compute nodes to detect and track blob-filaments in real time in fusion plasma. On a set of 30GB fusion simulation data, we observed linear speedup on 1024 processes and completed blob detection in less than three milliseconds using Edison, a Cray XC30 system at NERSC.Comment: 14 pages, 40 figure

Probing for the Trace Estimation of a Permuted Matrix Inverse Corresponding to a Lattice Displacement

In this work, we study probing for the more general problem of computing the trace of a permutation of $A^{-1}$, say $PA^{-1}$. The motivation comes from Lattice QCD where we need to construct "disconnected diagrams" to extract flavor-separated Generalized Parton functions. In Lattice QCD, where the matrix has a 4D toroidal lattice structure, these non-local operators correspond to a $PA^{-1}$ where $P$ is the permutation relating to some displacement $\vec{p}$ in one or more dimensions. We focus on a single dimension displacement ($p$) but our methods are general. We show that probing on $A^k$ or $(PA)^k$ do not annihilate the largest magnitude elements. To resolve this issue, our displacement-based probing works on $PA^k$ using a new coloring scheme that works directly on appropriately displaced neighborhoods on the lattice. We prove lower bounds on the number of colors needed, and study the effect of this scheme on variance reduction, both theoretically and experimentally on a real-world Lattice QCD calculation. We achieve orders of magnitude speedup over the unprobed or the naively probed methods